Magnetism in zero dimensions physics of magnetic molecules

Magnetism in zero dimensions physics of magnetic molecules Jürgen Schnack Department of Physics University of Bielefeld Germany http://obelix.physik.uni-bielefeld.de/ schnack/ Colloquium, Magdeburg University Magdeburg, December 16, 2009 unilogo-m-rot.jpg

? Collaboration Many thanks to my collaborators worldwide T. Englisch, T. Glaser, M. Höck, S. Leiding, A. Müller, R. Schnalle, Chr. Schröder, J. Ummethum (Bielefeld) K. Bärwinkel, H.-J. Schmidt, M. Neumann (Osnabrück); M. Luban, D. Vaknin (Ames Lab, USA); P. Kögerler (Aachen, Jülich, Ames) J. Musfeld (U. of Tennessee, USA); N. Dalal (Florida State, USA); R.E.P. Winpenny, E.J.L. McInnes (Man U, UK); L. Cronin (Glasgow, UK); J. van Slageren (Nottingham); H. Nojiri (Sendai, Japan); A. Postnikov (Metz, France) J. Richter, J. Schulenburg (Magdeburg); S. Blügel (FZ Jülich); A. Honecker (Göttingen); U. Kortz (Bremen); A. Tennant, B. Lake (HMI Berlin); B. Büchner, V. Kataev, R. Klingeler, H.-H. Klauß (Dresden); P. Chaudhuri (Mühlheim); J. Wosnitza (Dresden-Rossendorf) unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 1/51

? Contents for you today Contents for you today 1. Single Molecule Magnets 2. Antiferromagnetic Molecules 3. Molecules on Surfaces 4. Coherence Phenomena 5. Up to date theory modeling Fe 10 6. Frustration effects unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 2/51

? The beauty of magnetic molecules I The beauty of magnetic molecules I Inorganic or organic macro molecules, where paramagnetic ions such as Iron (Fe), Chromium (Cr), Copper (Cu), Nickel (Ni), Vanadium (V), Manganese (Mn), or rare earth ions are embedded in a host matrix; Pure organic magnetic molecules: magnetic coupling between high spin units (e.g. free radicals); Mn 12 Speculative applications: magnetic storage devices, magnets in biological systems, lightinduced nano switches, displays, catalysts, transparent magnets, qubits for quantum computers. unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 3/51

? The beauty of magnetic molecules II The beauty of magnetic molecules II Dimers (Fe 2 ), tetrahedra (Cr 4 ), cubes (Cr 8 ); Rings, especially iron and chromium rings Complex structures (Mn 12 ) drosophila of molecular magnetism; Soccer balls, more precisely icosidodecahedra (Fe 30 ) and other macro molecules; Chain like and planar structures of interlinked magnetic molecules, e.g. triangular Cu chain: J. Schnack, H. Nojiri, P. Kögerler, G. J. T. Cooper, L. Cronin, Phys. Rev. B 70, 174420 (2004); Sato, Sakai, Läuchli, Mila,... unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 4/51

? Single Molecule Magnets Single Molecule Magnets unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 5/51

? Single Molecule Magnets I Single Molecule Magnets I Magnetic Molecules may possess a large ground state spin, e.g. S = 10 for Mn 12 or Fe 8 ; Ground state spin can be stabilized by anisotropy (easy axis). unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 6/51

? Single Molecule Magnets II Single Molecule Magnets II Energy Anisotropy barrier Single Molecule Magnets (SMM): large ground state moment; anisotropy barrier dominates at low T. Orientation Magnetization H DS 2 z Metastable magnetization and hysteresis; Field But also magnetization tunneling due to noncommuting terms, e.g. E, B x, B y. H DS 2 z + E(S 2 x S 2 y) unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 7/51

? Single Molecule Magnets III Single Molecule Magnets III [Mn III 6 O 2 (Et-sao) 6 (O 2 CPh(Me 2 )) 2 (EtOH) 6 ]: S = 12 ground state with D = 0.43 cm 1 U eff = 86.4 K and a blocking temperature of about 4.5 K. A record molecule from the group of Euan Brechin (Edinburgh). C. J. Milios et al., J. Am. Chem. Soc. 129, 2754 (2007) S. Carretta et al., Phys. Rev. Lett. 100, 157203 (2008) unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 8/51

? Single Molecule Magnets IV Single Molecule Magnets IV Magnitude of the anisotropy barrier is mainly determined by the strength of the spin-orbit coupling and cannot be engineered by independently optimizing D and S. (1) From this point of view systems with larger energy barriers should be obtained in the case of perfect alignment of the Jahn-Teller axes... However, the challenge here will be the control of the ferromagnetic exchange. (1)... the widely considered design rule to increase S is not as efficient as suggested by H = DS 2,... the increase is on the order of unity and not S 2. (2) For obtaining better SMMs, it hence seems most promising to work on the local ZFS tensors D i or to work in a limit where the Heisenberg term is not dominant (i.e., to break the strong-exchange limit). (2) (1) E. Ruiz et al., Chem. Commun. 52 (2008). (2) O. Waldmann, Inorg. Chem. 46, 10035 (2007). unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 9/51

? Single Molecule Magnets V Single Molecule Magnets V Rational design of strict C 3 symmetry: Idea of Thorsten Glaser: C 3 symmetric alignment of local easy axes (easy axis Jahn-Teller axis); Various ions could be used so far, e.g. Mn 6 Cr (1), Mn 6 Fe (2),... Advantage: no E-terms, i.e. no (less) tunneling; Problem: exchange interaction sometimes antiferromagnetic. T. Glaser et al., Angew. Chem.-Int. Edit. 45, 6033 (2006). T. Glaser et al., Inorg. Chem. 48, 607 (2009). unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 10/51

? Antiferromagnetic Molecules Antiferromagnetic Molecules unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 11/51

? Antiferromagnetic Molecules I Antiferromagnetic Molecules I Rings To date: many AF rings synthesized, e.g. Fe 6, Fe 10, Fe 12,..., Cr 8,... (1) Theory: Exact diagonalization; Rotational band model; QMC; Classical (2) (1) Taft, Delfs, Saalfrank, Rentschler, Winpenny, Timco, Timco, Timco,... (2) Luban, Waldmann, Schnack, Schröder, Carretta, Engelhardt,... unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 12/51

? Antiferromagnetic Molecules II Antiferromagnetic Molecules II Trend A Investigation of spin dynamics and coherent tunnelling Tunneling of the Neel vector at low lemperatures (1,2,3); Not very practical, no easy observable, since S = 0; Tunneling in doped af rings (4). from (4) (1) O. Waldmann, Europhys. Lett. 60, 302 (2002). (2) A. Honecker, F. Meier, D. Loss, and B. Normand, Eur. Phys. J. B 27, 487 (2002). (3) F. Meier and D. Loss, Phys. Rev. Lett. 86, 5373 (2001). (4) F. Meier and D. Loss, Physica B 329-333, 1140 (2003). unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 13/51

? Antiferromagnetic Molecules III Antiferromagnetic Molecules III Trend B Synthesis of odd or heterometallic or coupled af spin rings Odd membered rings very rare; one reason: steric hindrance (1); Heterometallic rings derived from homometallic, especially from Cr 8 (2); Cr 7 Ni by Timco Coupling of heterometallic rings for quantum computing (3). (1) O. Cador et al., Angew. Chem. Int. Edit. 43, 5196 (2004); H. C. Yao et al., Chem. Commun. 1745 (2006); (2) F. K. Larsen et al., Angew. Chem. Int. Ed. 42, 101 (2003); E. Micotti et al., Phys. Rev. Lett. 97, 267204 (2006); L. P. Engelhardt et al., Angew. Chem. Int. Edit. 47, 924 (2008), i.e. Timco, Timco, Timco,... ; (3) G. A. Timco et al., Nature Nanotechnology (2009), accepted. unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 14/51

? Antiferromagnetic Molecules IV Antiferromagnetic Molecules IV Trend C Soliton dynamics Theoretical realization of classical solitons on af Heisenberg spin rings (1) Do quantum solitons exist and if, how do they look like? (2) (1) H.-J. Schmidt, C. Schröder, and M. Luban, cond-mat/0801.4262. (2) J. Schnack and P. Shchelokovskyy, J. Magn. Magn. Mater. 306, 79 (2006). unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 15/51

? Molecules on Surfaces Molecules on Surfaces unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 16/51

? Molecules on Surfaces I Molecules on Surfaces I Early attempts by Paul Müller (Erlangen) Cu 20 on Highly Orientated Pyrolytic Graphite (HOPG) (1); Scanning tunnelling microscopy (STM) (2); Scanning tunnelling spectroscopy (STS) (2); Current induced tunnelling spectroscopy (CITS) (2). from (1) Theory: Schoeller, Wegewijs, Timm, Postnikov, Kortus. (1) M. S. Alam et al., Inorg. Chem. 45, 2866 (2006). (2) M. Ruben, J. M. Lehn, and P. Müller, Chem. Soc. Rev. 35, 1056 (2006). unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 17/51

? Molecules on Surfaces II Molecules on Surfaces II Rings on surfaces Sulfur-functionalized clusters Cr 7 Ni on gold (1); Deposited from the liquid phase on Au(111); Scanning tunneling microscopy (STM) and X-ray photoemission spectroscopy (XPS); from (1) The stoichiometric behavior of the core level intensities, which are the direct fingerprint of the ring, confirms that the ring integrity is preserved. (1) (1) V. Corradini et al., Inorg. Chem. 46, 4937 (2007). (2006). unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 18/51

? Molecules on Surfaces III Molecules on Surfaces III Spin-polarized measurements Cobalt-phthalocyanine molecules on cobalt islands (1); Spin-polarized STM and STS; Transport through polarized Co islands; from (1) Identification of ferromagnetic molecule-lead exchange interaction (1) (1) C. Iacovita et al., Physical Review Letters 101, 116602 (2008). unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 19/51

? Molecules on Surfaces IV Molecules on Surfaces IV from (1) Backslash on molecule How much of the deposited molecule survives? Study of a Mn 6 cluster grafted on a Polyoxometalate (POM) (1); Intra-molecular interactions change compared to free molecule (1). (1) Xikui Fang and P. Kögerler, Chem. Commun. 3396 (2008). unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 20/51

? Molecules on Surfaces V Molecules on Surfaces V Carbon nanotube squid Use of single-walled carbon nanotube (CNT) Josephson junctions; Discrete quantum dot (QD) energy level structure controlled by gates (1); from (1) CNT-SQUIDs sensitive local magnetometers to study the magnetization reversal of individual magnetic particles (1). (1) J. P. Cleuziou et al., Nature Nanotechnology 1, 53 (2006). unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 21/51

? Coherence Phenomena Coherence Phenomena unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 22/51

? Coherence Phenomena I Coherence Phenomena I Quantum computing Chemical realization through coupled molecules with switchable coupling; Original ideas, see e.g. (2); Molecular transistors; transport in weak or strong coupling regime (3). from (1) Needed: long coherence times. (1) G. A. Timco et al., Nature Nanotechnology (2009), accepted; R. E. P. Winpenny, Angew. Chem. Int. Ed. 47, 7992 (2008); M. Affronte et al., Dalton Transactions 2810 (2006); M. Affronte et al., J. Magn. Magn. Mater. 310, E501 (2007). (2) M. N. Leuenberger and D. Loss, Nature 410, 789 (2001). (3) L. Bogani and W. Wernsdorfer, Nature Materials 7, 179 (2008). unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 23/51

? Coherence Phenomena II Coherence Phenomena II Spin relaxation times EPR/NMR, Hahn echo techniques, T 1, T 2 times; Decoherence due to e.g. nuclei, phonons, dipolar interaction; Deuteration improves coherence times considerably; from (1) µs (!) can be reached. (1) (1) A. Ardavan et al., Phys. Rev. Lett. 98, 057201 (2007). (2) S. Bahr, K. Petukhov, V. Mosser, and W. Wernsdorfer, Phys. Rev. Lett. 99, 147205 (2007); W. Wernsdorfer, Nature Materials 6, 174 (2007). (3) S. Bertaina et al., Nature 453, 203 (2008). (4) C. Schlegel et al., Phys. Rev. Lett. 101, 147203 (2008). unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 24/51

? Up to date theory modeling Up to date theory modeling unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 25/51

? Advanced ITO & Point Groups I Advanced ITO & Point Groups I Group theory for highly symmetric molecules: H = i,j J ij s i s j + gµ B S B ; [ H, S ] 2 = 0, [ ] H, S z = 0; Irreducible Tensor Operator (ITO) approach, MAGPACK (1); Additional point group symmetries (2). (1) D. Gatteschi and L. Pardi, Gazz. Chim. Ital. 123, 231 (1993); J. J. Borras-Almenar, J. M. Clemente-Juan, E. Coronado, and B. S. Tsukerblat, Inorg. Chem. 38, 6081 (1999). (2) O. Waldmann, Phys. Rev. B 61, 6138 (2000); V. E. Sinitsyn, I. G. Bostrem, and A. S. Ovchinnikov, J. Phys. A-Math. Theor. 40, 645 (2007); R. Schnalle and J. Schnack, Phys. Rev. B 79, 104419 (2009). unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 26/51

? Reminder ITO Reminder ITO H Heisenberg = 3 i,j J ij T (0) ({k i }, {k i } k i = k j = 1) Irreducible Tensor Operator approach Express spin operators and functions thereof as ITOs; Use vector coupling basis α S M and recursive recoupling; Numerical implementation e.g. MAGPACK. (1) Gatteschi, Tsukerblat, Coronado, Waldmann,... (2) R. Schnalle, Ph.D. thesis, Osnabrück University (2009) unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 27/51

? Advanced ITO & Point Groups II Advanced ITO & Point Groups II P (n) α S M = ( l n h R ) ( χ (R)) (n) G (R) α S M Point Group Symmetry Projection on irreducible representations (Wigner); Basis function generating machine; Orthonormalization necessary. (1) O. Waldmann, Phys. Rev. B 61, 6138 (2000). (2) R. Schnalle, Ph.D. thesis, Osnabrück University (2009) unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 28/51

? Advanced ITO & Point Groups III Advanced ITO & Point Groups III G (R) α S M a = α α S M a a α S M α S M b Serious problem: Recoupling So far: only point groups that are compatible with the coupling scheme are used (1); Problem: otherwise complicated basis transformation between different coupling schemes; Solution: implementation of graph-theoretical results to evaluate recoupling coefficients a α S M α S M b (2). (1) O. Waldmann, Phys. Rev. B 61, 6138 (2000). (2) R. Schnalle, Ph.D. thesis, Osnabrück University (2009) unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 29/51

? Advanced ITO & Point Groups IV Advanced ITO & Point Groups IV E/ J -31-32 -33-34 A1g A2g Eg T1g T2g A1u A2u Eu T1u T2u -35-36 0 1 2 3 S Cuboctahedron, s = 3/2, Hilbert space dimension 16,777,216; symmetry O h (1). Evaluation of recoupling coefficients very time consuming. (1,2) (1) J. Schnack and R. Schnalle, Polyhedron 28, 1620 (2009); (2) R. Schnalle and J. Schnack, Phys. Rev. B 79, 104419 (2009). unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 30/51

? Anisotropic magnetic molecules I Anisotropic magnetic molecules I Theory H ( B) = i,j J ij s (i) s (j) + i d i ( ei s (i) ) 2 + µb B N g i s (i) i [ H, S ] 2 0, [ H, S ] z 0; You have to diagonalize H ( B) for every field (direction and strength)! Orientational average. If you are lucky, point group symmetries still exist. Use them! Easy: dim(h) < 30, 000; possible: 30, 000 < dim(h) < 140, 000 T. Glaser et al. et J. Schnack, Inorg. Chem. 48, 607 (2009). unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 31/51

? Anisotropic magnetic molecules II Anisotropic magnetic molecules II Example C 3 What can be achieved? Mn 3 Cr: Mn Cr e Two couplings: J 1 to central Cr, J 2 between Mn; Mn: s=5/2, g=2.0; Cr: s=3/2, g=1.95 Mn Mn Model Mn anisotropy by local axis e(ϑ, φ). Due to C 3 symmetry ϑ Mn1 = ϑ Mn2 = ϑ Mn3. Only relative φ = 120 determined. Model Cr anisotropy by local axis e(ϑ, φ). Due to C 3 symmetry ϑ Cr = 0, φ Cr = 0. Result: J 1 = 0.29 cm 1, J 2 = 0.08 cm 1, d Mn = 1.21 cm 1, ϑ Mn = 22, d Cr = +0.17 cm 1. ab initio calculations need. unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 32/51

? Frustration effects Frustration effects unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 33/51

? Definition of frustration Definition of frustration Simple: An antiferromagnet is frustrated if in the ground state of the corresponding classical spin system not all interactions can be minimized simultaneously. A B A B Advanced: A non-bipartite antiferromagnet is frustrated. A bipartite spin system can be decomposed into two sublattices A and B such that for all exchange couplings: J(x A, y B ) g 2, J(x A, y A ) g 2, J(x B, y B ) g 2, cmp. (1,2). B A A B B A A B (1) E.H. Lieb, T.D. Schultz, and D.C. Mattis, Ann. Phys. (N.Y.) 16, 407 (1961) (2) E.H. Lieb and D.C. Mattis, J. Math. Phys. 3, 749 (1962) B A B A unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 34/51

? Marshall-Peierls sign rule for even rings Marshall-Peierls sign rule for even rings Expanding the ground state in H(M) in the product basis yields a sign rule for the coefficients Ψ 0 = m c( m) m with N m i = M i=1 Ns c( m) = ( 1) 2 P N/2 i=1 m 2i a( m) All a(m) are non-zero, real, and of equal sign. Yields eigenvalues for the shift operator T : exp { } i 2πk N with k a N 2 mod N, a = Ns M (1) W. Marshall, Proc. Royal. Soc. A (London) 232, 48 (1955) unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 35/51

? Numerical findings for odd rings Numerical findings for odd rings For odd N and half integer s, i.e. s = 1/2, 3/2, 5/2,... we find that (1) - the ground state has total spin S = 1/2; - the ground state energy is fourfold degenerate. Reason: In addition to the (trivial) degeneracy due to M = ±1/2, a degeneracy with respect to k appears (2): k = N+1 4 and k = N N+1 4 For the first excited state similar rules could be numerically established (3). (1) K. Bärwinkel, H.-J. Schmidt, J. Schnack, J. Magn. Magn. Mater. 220, 227 (2000) (2) largest integer, smaller or equal (3) J. Schnack, Phys. Rev. B 62, 14855 (2000) unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 36/51

? k-rule for odd rings k-rule for odd rings An extended k-rule can be inferred from our numerical investigations which yields the k quantum number for relative ground states of subspaces H(M) for even as well as odd spin rings, i.e. for all rings (1) N k ±a mod N, a = Ns M 2 k is independent of s for a given N and a. The degeneracy is minimal (N 3). a N s 0 1 2 3 4 5 6 7 8 9 8 1/2 0 4 8 0 12 4 16 0 - - - - - 9 1/2 0 5 4 10 1 15 3 20 2 - - - - - 9 1 0 5 4 10 1 15 3 20 2 25 2 30 3 35 1 40 4 45 0 No general, but partial proof yet. (1) K. Bärwinkel, P. Hage, H.-J. Schmidt, and J. Schnack, Phys. Rev. B 68, 054422 (2003) unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 37/51

? Fe 30 and friends Fe 30 and friends Corner sharing triangles and tetrahedra + + + Several frustrated antiferromagnets show an unusual magnetization behavior, e.g. plateaus and jumps. Example systems: icosidodecahedron, kagome lattice, pyrochlore lattice. unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 38/51

? Rotational bands in non-frustrated antiferromagnets Rotational bands in non-frustrated antiferromagnets φ B A B A B A Often minimal energies E min (S) form a rotational band: Landé interval rule (1); For bipartite systems (2,3): H eff = 2 J eff S A S B; Lowest band rotation of Néel vector, second band spin wave excitations (4). (1) A. Caneschi et al., Chem. Eur. J. 2, 1379 (1996), G. L. Abbati et al., Inorg. Chim. Acta 297, 291 (2000) (2) J. Schnack and M. Luban, Phys. Rev. B 63, 014418 (2001) (3) O. Waldmann, Phys. Rev. B 65, 024424 (2002) (4) P.W. Anderson, Phys. Rev. B 86, 694 (1952), O. Waldmann et al., Phys. Rev. Lett. 91, 237202 (2003). unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 39/51

? Giant magnetization jumps in frustrated antiferromagnets I Giant magnetization jumps in frustrated antiferromagnets I {Mo 72 Fe 30 } Close look: E min (S) linear in S for high S instead of being quadratic (1); Heisenberg model: property depends only on the structure but not on s (2); Alternative formulation: independent localized magnons (3); (1) J. Schnack, H.-J. Schmidt, J. Richter, J. Schulenburg, Eur. Phys. J. B 24, 475 (2001) (2) H.-J. Schmidt, J. Phys. A: Math. Gen. 35, 6545 (2002) (3) J. Schulenburg, A. Honecker, J. Schnack, J. Richter, H.-J. Schmidt, Phys. Rev. Lett. 88, 167207 (2002) unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 40/51

? Giant magnetization jumps in frustrated antiferromagnets II Giant magnetization jumps in frustrated antiferromagnets II Localized Magnons 5 localized magnon = 1 2 ( 1 2 + 3 4 ) 1 2 1 = s (1)... etc. 8 4 3 6 H localized magnon localized magnon 7 Localized magnon is state of lowest energy (1,2). Triangles trap the localized magnon, amplitudes cancel at outer vertices. (1) J. Schnack, H.-J. Schmidt, J. Richter, J. Schulenburg, Eur. Phys. J. B 24, 475 (2001) (2) H.-J. Schmidt, J. Phys. A: Math. Gen. 35, 6545 (2002) unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 41/51

? Giant magnetization jumps in frustrated antiferromagnets III Giant magnetization jumps in frustrated antiferromagnets III 10 11 6 1 2 5 7 Non-interacting one-magnon states can be placed on various molecules, e. g. 2 on the cuboctahedron and 3 on the icosidodecahedron (3rd delocalized); 9 4 8 3 12 Each state of n independent magnons is the ground state in the Hilbert subspace with M = Ns n; 29 24 30 25 26 17 19 20 18 10 6 16 9 1 5 2 4 3 8 7 11 13 12 21 27 Linear dependence of E min on M (T = 0) magnetization jump; A rare example of analytically known many-body states! 23 15 28 14 22 J. Schnack, H.-J. Schmidt, J. Richter, J. Schulenburg, Eur. Phys. J. B 24, 475 (2001) unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 42/51

? Giant magnetization jumps in frustrated antiferromagnets III Giant magnetization jumps in frustrated antiferromagnets III Kagome Lattice + + + Non-interacting one-magnon states can be placed on various lattices, e. g. kagome or pyrochlore; Each state of n independent magnons is the ground state in the Hilbert subspace with M = Ns n; Kagome: max. number of indep. magnons is N/9; 1 0.8 7/9 Linear dependence of E min on M (T = 0) magnetization jump; m(h) 0.6 0.4 2/3 2.6 3 Jump is a macroscopic quantum effect! 0.2 N=27 36 45 54 0 0 1 2 3 h A rare example of analytically known many-body states! J. Schulenburg, A. Honecker, J. Schnack, J. Richter, H.-J. Schmidt, Phys. Rev. Lett. 88, 167207 (2002) J. Richter, J. Schulenburg, A. Honecker, J. Schnack, H.-J. Schmidt, J. Phys.: Condens. Matter 16, S779 (2004) unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 43/51

? Condensed matter physics point of view: Flat band Condensed matter physics point of view: Flat band 1 +1 +1 2 J 1 J 2 =2J 1 m(h) 0.8 0.6 0.4 sawtooth chain sawtooth chain 0.2 N=24 N=32 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 h Flat band of minimal energy in one-magnon space; localized magnons can be built from delocalized states in the flat band. Entropy can be evaluated using hard-object models (1); universal lowtemperature behavior. Same behavior for Hubbard model; flat band ferromagnetism (Tasaki & Mielke), jump of N with µ (2). (1) H.-J. Schmidt, J. Richter, R. Moessner, J. Phys. A: Math. Gen. 39, 10673 (2006) (2) A. Honecker, J. Richter, Condens. Matter Phys. 8, 813 (2005) unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 44/51

? Magnetocaloric effect I Magnetocaloric effect I Giant jumps to saturation Many Zeeman levels cross at one and the same magnetic field. E M M= 4 M= 3 M(T=0,B) You know this for a giant spin at B = 0. M= 2 M= 1 High degeneracy of ground state levels large residual entropy at T = 0. B ( ) T = T B S C ( ) S B T M. Evangelisti et al., Appl. Phys. Lett. 87, 072504 (2005). J. Schulenburg, A. Honecker, J. Schnack, J. Richter, H.-J. Schmidt, Phys. Rev. Lett. 88, 167207 (2002) M. E. Zhitomirsky, Phys. Rev. B 67, 104421 (2003). M. E. Zhitomirsky and A. Honecker, J. Stat. Mech.: Theor. Exp. 2004, P07012 (2004). unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 45/51

? Magnetocaloric effect II Magnetocaloric effect II Isentrops of af s = 1/2 dimer 2 T-B isentropes Magnetocaloric effect: 1.5 a b (a) reduced, (b) the same, 1 (c) enhanced, 0.5 c (d) opposite when compared to an ideal paramagnet. 0 0 0.5 1 1.5 2 2.5 3 Case (d) does not occur for a paramagnet. blue lines: ideal paramagnet, red curves: af dimer unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 46/51

? Magnetocaloric effect III Magnetocaloric effect III Molecular systems Cuboctahedron: high cooling rate due to independent magnons; Ring: normal level crossing, normal jump; Icosahedron: unusual behavior due to edge-sharing triangles, high degeneracies all over the spectrum; high cooling rate. J. Schnack, R. Schmidt, J. Richter, Phys. Rev. B 76, 054413 (2007) unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 47/51

? Metamagnetic phase transition I Metamagnetic phase transition I Hysteresis without anisotropy M/M sat 0.5 0.4 0.3 0.2 0.1 up cycle (simulation) down cycle (simulation) analytical results T = 0 Heisenberg model with isotropic nearest neighbor exchange Hysteresis behavior of the classical icosahedron in an applied magnetic field. Classical spin dynamics simulations (thick lines). 0 0 0.1 0.2 0.3 0.4 0.5 B/B sat Analytical stability analysis (grey lines). Movie. C. Schröder, H.-J. Schmidt, J. Schnack, M. Luban, Phys. Rev. Lett. 94, 207203 (2005) unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 48/51

? Metamagnetic phase transition III Metamagnetic phase transition III Quantum icosahedron Quantum analog: Non-convex minimal energy levels magnetization jump of M > 1. Lanczos diagonalization for various s vectors with up to 10 9 entries. True jump of M = 2 for s = 4. Polynomial fit in 1/s yields the classically observed transition field. C. Schröder, H.-J. Schmidt, J. Schnack, M. Luban, Phys. Rev. Lett. 94, 207203 (2005) unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 49/51

? The End Thank you very much for your attention. unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 50/51

? Information Molecular Magnetism Web www.molmag.de Highlights. Tutorials. Who is who. Conferences. unilogo-m-rot.jpg Jürgen Schnack, Magnetism in zero dimensions 51/51